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### Open Access Journals

CPAA

We study nonlinear Dirichlet problems driven by the scalar
$p$-Laplacian with a nonsmooth potential. First for the so-called
"sublinear problem", under nonuniform nonresonance conditions,
we establish the existence of at least one strictly positive
solution. Then we prove two multiplicity results for positive
solutions. The first concerns the "superlinear problem" and the
second is for the sublinear problem. The method of proof is
variational based on the nonsmooth critical point theory for
locally Lipschitz functions. Our results complement the ones
obtained by De Coster (Nonlin.Anal.23 (1995)).

CPAA

We consider a nonlinear elliptic equation of logistic
type, driven by the $p$-Laplacian differential operator with a
general superdiffusive reaction. We show that the equation
exhibits a bifurcation phenomenon. Namely there is a critical
value $\lambda_*$ of the parameter $\lambda>0$, such that, if
$\lambda>\lambda_*$, the equation has two nontrivial positive
smooth solutions, if $\lambda=\lambda_*,$ then there is one
positive solution and finally if $\lambda\in (0,\lambda_*),$ then
there is no positive solution.

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